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Question

Prove that:

tan 8212=(3+2)(2+1)=2+3+4+6

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Solution

tan 8212=tan (90712)=cot 712=cot A If A=712

Now

cot A=cos Asin A=2 cos2 A2 sin A cos A=1+cos2 Asin2 Acot A=1+cos 15sin 15=1+cos(4530)sin 15

=1+(12×32+12×12)12×3212×12=22+(3+1)31×3+13+1=22(3+1)+(3+1)231=26+22+4+232cot A=6+2+2+3 (i)=2+2+6+3

=2+(1+2)+3(2+1)cot A=(2+1)(2+3) (ii)

From equation (i) and (ii)

tan 8212=cot 712=2+3+4+6=(2+1)(2+3)


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